Nonlinear singular integral inequalities for functions in two and \(n\) independent variables
The author establishes a priori bounds on solutions of some nonlinear integral inequalities with weakly singular kernels for functions in two and \(n\) independent variables. The typical inequality considered is of the form \[ u(x,y)\leq a(x,y)+ \int^x_0 \int^y_0 (x- s)^{a-1}(y- t)^{\beta- 1}F(s,t) \omega(u(s,t)) ds dt,\;x,y\in [0,T),\tag{\(*\)} \] where \(0< T\leq \infty\), \(\alpha> 0\), \(\beta> 0\), \(u,F:[0,T)^2\to R'= [0,\infty)\) are continuous functions, \(\omega: R^+\to R^+\) is a \(C^1\)-function, and \(a(x,y): [0,T)^2\to R'\) is a \(C^2\)-function satisfying the condition \(a_{xy}(x,y)\geq 0\), \(a_x(x,y)\geq 0\) [or \(a_y(x,y)\geq 0\)]. In Theorem 2.2 (the main result), a priori bounds on solutions of inequality \((*)\) are obtained for two cases: (i) \(\alpha>{1\over 2}\), \(\beta>{1\over 2}\) and \(\omega\) satisfies the condition \((q)\) with \(q= 2\); and (ii) \(\alpha= \beta= 1/(z+ 1)\) and \(\omega\) satisfies the condition (q) with \(q= z+ 2\), where \(z\geq 1\) is a real number. Definition. Let \(q> 0\) be a real number. A function \(\omega: R^+\to R^+\) is called satisfying a condition (q) if \[ e^{-qt}[\omega(u)]^q\leq R(t) \omega(e^{-qt} u^q),\quad u\in R^+,\quad 0\leq t< T, \] holds for some continuous function \(R(t): R^+\to R^+\). The argument used herein is an approach proposed by the present author in [J. Math. Anal. Appl. 214, No. 2, 349-366 (1997; Zbl 0893.26006)]. A singular integral inequality for \(n\)-independent-variable functions is also given.
scientific article; zbMATH DE number 1507756